Finite Group Databasebeta

← #10690519

Subgroups — #10690519

Order 768.

726 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C22⋊Q8C211yesmaximal, characteristic
2C22×C4×C3⋊C8C211yesmaximal, characteristic
3C2×#4535C212yesmaximal, complemented
4#71530C211yesmaximal, characteristic
5#67619C212yesmaximal, complemented
6#9071—31nomaximal, Sylow
7C2×C3×C22⋊Q8C411yescharacteristic
8C2×C3×C4×Q8C2211yescharacteristic
9C22×C3×C42C2211yescharacteristic
10C23×C3⋊C8—21no—
11C3×#391C411yescharacteristic
12C3×#389C411yescharacteristic
13C3×#387—21no—
14C3×#385—21no—
15C3×#380C411yescharacteristic
16C3×C4×C4⋊C4—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
18C3×C4×C22⋊C4—21no—
19C2×#830—21no—
20C2×C4⋊(C3⋊C8)C2212yes—
21C2×C4×C3⋊C8—21no—
22C2×C4×C3⋊C8—24no—
23C2×C4×C3⋊C8C2212yes—
24#4535—24no—
25#4524—21no—
26C22×C4×C8—31no—
27C4×C22⋊Q8D311yescharacteristic
28#1797—31no—
29C2×Q8⋊C8—32no—
30#1279—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C23×C3×C4D411yescharacteristic
35C3×C22⋊Q8—22no—
36C3×C22⋊Q8C814yescomplemented
37C3×C4×Q8—22no—
38C3×C4×Q8—22no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C42—22no—
48C2×C3×C42—22no—
49C2×C3×C42—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2×C411yescharacteristic
52C2×C3×C42D411yescharacteristic
53C2×C3×C42D411yescharacteristic
54C2×C3×C42C2311yescharacteristic
55C22×C3⋊C8—24no—
56C22×C3⋊C8—24no—
57C22×C3⋊C8—24no—
58C22×C3⋊C8—22no—
59C22×C3⋊C8—22no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95—21no—
64C3×#95—21no—
65#830—42no—
66C4⋊(C3⋊C8)—22no—
67C4⋊(C3⋊C8)—22no—
68C4×C3⋊C8—24no—
69C4×C3⋊C8—22no—
70C4×C3⋊C8D414yes—
71C23×C8—61no—
72C2×C22⋊Q8Dic311yescharacteristic
73C2×C4×Q8C2×D311yescharacteristic
74C22×C42C2×D311yescharacteristic
75C2×C4⋊C8—32no—
76C2×C22⋊C8—61no—
77C2×C4×C8—32no—
78C2×C4×C8—64no—
79C2×C4×C8—61no—
80#391Dic311yescharacteristic
81#389Dic311yescharacteristic
82#387—21no—
83#385—21no—
84#380Dic311yescharacteristic
85C4×C4⋊C4—21no—
86C4×C4⋊C4C2×D311yescharacteristic
87C4×C22⋊C4—21no—
88#336—61no—
89Q8⋊C8—64no—
90C24×C3C22⋊C411yescharacteristic
91C2×C3×Q8—21no—
92C2×C3×Q8—22no—
93C2×C3×Q8—22no—
94C2×C3×Q8C2×C812yescomplemented
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—21no—
99C22×C3×C4—22no—
100C22×C3×C4—21no—
101C22×C3×C4—22no—
102C22×C3×C4—22no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4C22⋊C411yescharacteristic
108C22×C3×C4C22⋊C411yescharacteristic
109C22×C3×C4C22⋊C411yescharacteristic
110C22×C3×C4C2×C812yes—
111C22×C3×C4M1612yes—
112C22×C3×C4SD1612yes—
113C22×C3×C4Q1612yes—
114C22×C3×C4C22×C411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C3×C4⋊C4—24no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—42no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4C2×C812yescomplemented
125C3×C22⋊C4—24no—
126C3×C22⋊C4—42no—
127C3×C42—22no—
128C3×C42—21no—
129C3×C42—22no—
130C3×C42—21no—
131C3×C42—42no—
132C3×C42—42no—
133C3×C42C2×D412yes—
134C3×C42#4112yes—
135C2×C3⋊C8—216no—
136C2×C3⋊C8—24no—
137C2×C3⋊C8—24no—
138C2×C3⋊C8—24no—
139C2×C3⋊C8—42no—
140C2×C3⋊C8—42no—
141C2×C3⋊C8—42no—
142C22×Q8C2×Dic311yescharacteristic
143C23×C4C2×Dic311yescharacteristic
144C23×C4#6711yescharacteristic
145C23×C4#6711yescharacteristic
146C22×C8—64no—
147C22×C8—64no—
148C22×C8—64no—
149C22×C8—62no—
150C22×C8—62no—
151C22⋊Q8—22no—
152C22⋊Q8C3⋊C814yescomplemented
153C4×Q8—22no—
154C4×Q8—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4C2×Dic311yescharacteristic
159C2×C4⋊C4C2×Dic311yescharacteristic
160C2×C4⋊C4C2×Dic311yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42C2×Dic311yescharacteristic
168C2×C42#6711yescharacteristic
169C2×C42#6711yescharacteristic
170C2×C42C22×D311yescharacteristic
171C4⋊C8—62no—
172C4⋊C8—62no—
173C22⋊C8—122no—
174C4×C8—34no—
175C4×C8—64no—
176C4×C8—62no—
177#95—21no—
178#95—21no—
179#95—21no—
180#95—21no—
181#95—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3C22⋊C812yes—
187C23×C3#10312yes—
188C23×C3#10412yes—
189C23×C3C2×C22⋊C411yescharacteristic
190C3×Q8—24no—
191C3×Q8—42no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—24no—
203C2×C3×C4—24no—
204C2×C3×C4—24no—
205C2×C3×C4—24no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4C22⋊C812yes—
219C2×C3×C4#10312yes—
220C2×C3×C4#10412yes—
221C2×C3×C4C4×D412yes—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12112yes—
224C2×C3×C4#12311yescharacteristic
225C2×C3×C4C22×C811yescharacteristic
226C2×C3×C4C2×M1611yescharacteristic
227C2×C3×C4#13112yes—
228C2×C3×C4C2×SD1611yescharacteristic
229C2×C3×C4C2×Q1611yescharacteristic
230C2×C3×C4#13512yes—
231C3⋊C8—28no—
232C3⋊C8—82no—
233C24#21711yescharacteristic
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8C2×C3⋊C812yescomplemented
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4C2×C3⋊C812yes—
251C22×C4#20812yes—
252C22×C4#21512yes—
253C22×C4#21612yes—
254C22×C4#21711yescharacteristic
255C22×C4#21711yescharacteristic
256C22×C4#21711yescharacteristic
257C22×C4C22×Dic311yescharacteristic
258C22×C4C2×#6711yescharacteristic
259C22×C4C2×#6711yescharacteristic
260C2×C8—616no—
261C2×C8—64no—
262C2×C8—64no—
263C2×C8—64no—
264C2×C8—122no—
265C2×C8—122no—
266C2×C8—122no—
267C4⋊C4—24no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4C2×C3⋊C812yescomplemented
275C22⋊C4—24no—
276C22⋊C4—42no—
277C42—22no—
278C42—21no—
279C42—21no—
280C42—22no—
281C42—42no—
282C42—42no—
283C42D3×D412yes—
284C42#23712yes—
285C22×C3—22no—
286C22×C3—22no—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3Q8⋊C814yescomplemented
292C22×C3#38611yescharacteristic
293C22×C3C2×C22⋊C811yescharacteristic
294C22×C3#40811yescharacteristic
295C22×C3C2×#10311yescharacteristic
296C22×C3#41611yescharacteristic
297C22×C3C2×#10411yescharacteristic
298C22×C3C2×#10411yescharacteristic
299C3×C4—22no—
300C3×C4—22no—
301C3×C4—21no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—42no—
305C3×C4—42no—
306C3×C4—81no—
307C3×C4—81no—
308C3×C4C8×D412yes—
309C3×C4#43512yes—
310C3×C4#46512yes—
311C3×C4#46712yes—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23#83012yes—
317C23#83512yes—
318C23#83712yes—
319C23C2×#21711yescharacteristic
320Q8—24no—
321Q8—42no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4#83012yes—
349C2×C4#83512yes—
350C2×C4#83712yes—
351C2×C4C22×C3⋊C811yescharacteristic
352C2×C4C2×#20811yescharacteristic
353C2×C4D4×Dic312yes—
354C2×C4#93511yescharacteristic
355C2×C4#93711yescharacteristic
356C2×C4#93912yes—
357C2×C4C2×#21511yescharacteristic
358C2×C4C2×#21611yescharacteristic
359C2×C4#94812yes—
360C2×C4#95012yes—
361C8—68no—
362C8—242no—
363C2×C3—24no—
364C2×C3C2×Q8⋊C811yescharacteristic
365C2×C3#147611yescharacteristic
366C2×C3#149311yescharacteristic
367C2×C3#149411yescharacteristic
368C2×C3#186911yescharacteristic
369C2×C3#189211yescharacteristic
370C2×C3#189611yescharacteristic
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22#453514yescomplemented
378C22C2×#83011yescharacteristic
379C22#522011yescharacteristic
380C22C2×#83511yescharacteristic
381C22#523311yescharacteristic
382C22#523411yescharacteristic
383C22C2×#83711yescharacteristic
384C22C2×#83711yescharacteristic
385C4—22no—
386C4—21no—
387C4—22no—
388C4—21no—
389C4—22no—
390C4—42no—
391C4—42no—
392C4—81no—
393C4—81no—
394C4D4×C3⋊C812yes—
395C4#501112yes—
396C4#504812yes—
397C4#505112yes—
398C3#907111yescharacteristic, complemented, Sylow
399C2—24no—
400C2#7161011yescharacteristic
401C2C2×#453511yescharacteristic
402C2#7163811yescharacteristic
403C2#7169611yescharacteristic
404C2#7171111yescharacteristic
405C2#7175211yescharacteristic
406C2#7182611yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×C4×C22⋊Q8C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenterC4×C22⋊Q8D3
hypocenterC3#9071
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#4535C2semidirect
5. #67619C2semidirect
36. C3×C22⋊Q8C8semidirect
94. C2×C3×Q8C2×C8semidirect
124. C3×C4⋊C4C2×C8semidirect
152. C22⋊Q8C3⋊C8semidirect
237. C2×Q8C2×C3⋊C8semidirect
274. C4⋊C4C2×C3⋊C8semidirect
291. C22×C3Q8⋊C8semidirect
377. C22#4535semidirect
398. C3#9071semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×C4×C22⋊Q82
C22×C4×C3⋊C82
C2×#45352
#715302
#676192
#90713